Lecture – 24

Calculus of Variations : An Overview

Dr. Radhakant PadhiAsst. Professor

Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

2

Fundamental Theorems of Calculus

Theorem – 1:

Theorem – 2:

Theorem – 3:

( ) ( ) ( ), provided is continuousx

a

d f d f x f xdx

σ σ =∫

( ) ( )

( ) ( )

,,

provided , has continuous partial derivative /

b b

a a

f x yd f x y dy dydx x

f x y f x

∂=

∂

∂ ∂

∫ ∫

( ) ( ) ( ) ( )2 2

1 1

( ) ( )2 1

2 1( ) ( )

1 2

,, , ( ) , ( )

provided , , have continuous partial derivatives with respect to

x x

x x

f x y d dd f x y dy dy f x x f x xdx x dx dx

f x

ψ ψ

ψ ψ

ψ ψψ ψ

ψ ψ

∂ ⎡ ⎤= + −⎢ ⎥∂ ⎣ ⎦∫ ∫

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

3

Calculus of Variations:Basic Concepts

Function (to each value of the independent variable, there is a corresponding value of the dependent variable)

Increment of a function

Functional (to each function, there is a corresponding value of the dependent variable)

Increment of a functional

( ) 32 3x t t t= + ( )( ) ( )

( )

1

01

3

0

2 3 2

J x t x t dt

t t dt

=

= + =

∫

∫

( ) ( )x x t t x tΔ + Δ − ( ) ( )( ) ( )( )J J x t x t J x tδΔ + −

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

4

Example

( ) ( )

( )

( )( )

( )( )( )( )

0 0

0

0

0

2 2

2 2

22 2

2

( ) ( ) ( )

2 ( ) ( ) 1 2 ( ) 1

2 ( ) ( ) 1 2 ( ) 1

2 ( ) 2 ( ) ( ) ( ) 1 2 ( ) 1

4 ( ) ( ) 2 ( )

f f

f

f

f

t t

t t

t

t

t

t

t

t

J J x t x t J x t

x t x t dt x t dt

x t x t x t dt

x t x t x t x t x t dt

x t x t x t dt

δ

δ

δ

δ δ

δ δ

Δ = + −

⎡ ⎤ ⎡ ⎤= + + − +⎣ ⎦⎣ ⎦

⎡ ⎤ ⎡ ⎤= + + − +⎣ ⎦⎣ ⎦

⎡ ⎤ ⎡ ⎤= + + + − +⎣ ⎦⎣ ⎦

⎡ ⎤= +⎣ ⎦

∫ ∫

∫

∫

∫

0

22 ( ) 1ft

t

J x t dt⎡ ⎤= +⎣ ⎦∫

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Calculus of Variations:Basic Concepts

Function and its increment Functional and its increment

Reference: D. S. Naidu, Optimal Control Systems, CRC Press, 2002.

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Calculus of Variations:Basic Concepts

Differential of a function Variation of a functional

( ) ( )

( )

( )

* *

2

*

* * *

22

2

: First deviation : Second deviation

2

* *

0 0

12!

lim lim

. . in general

t t

df d f

t tt

f f t t f t

df d ft tdt dt

df d f

dfdf f tdt

i e df f t

Δ → Δ →

Δ = + Δ −

⎛ ⎞⎛ ⎞= Δ + Δ +⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= + +

⎛ ⎞= Δ = Δ⎜ ⎟

⎝ ⎠

= Δ

( ) ( )

( )2

22

2

: First variation : Second variation

2

( ) ( ) ( )

12!

J J

J J x t x t J x t

J Jx xx x

J J

JJ xx

δ δ

δ

δ δ

δ δ

δ δ

Δ = + −

⎛ ⎞∂ ∂⎛ ⎞= + +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

= + +

∂⎛ ⎞= ⎜ ⎟∂⎝ ⎠

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

7

Calculus of Variations:Basic Concepts

Result – 1:Derivative of variation = Variation of derivative

Result – 2:Integration of variation = Variation of integration

( ) ( ) ( )

( ) ( )

( )

*

*

d dx t x t x tdt dt

dx t dx tdt dt

x t

δ

δ

⎡ ⎤⎡ ⎤ = −⎣ ⎦ ⎣ ⎦

= −

⎡ ⎤= ⎣ ⎦

( ) ( ) ( )

( ) ( )

( )

0 0

0 0

0

*

*

t t

t t

t t

t t

t

t

x d x x d

x d x d

x d

δ τ τ τ τ τ

τ τ τ τ

δ τ τ

⎡ ⎤= −⎣ ⎦

= −

⎡ ⎤= ⎢ ⎥

⎢ ⎥⎣ ⎦

∫ ∫

∫ ∫

∫

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Exercise

( )0

2

Evaluate the variation of :

( ) 2 ( ) 3 ( ) 4

Note: By 'variation', we mean 'first variation' (by default)

ft

t

J x t x t x t dt⎡ ⎤= + +⎣ ⎦∫

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Solution

( ) ( )

[ ]

0 0

0

0

0

2 2

2 2 2

2

:( ) ( ) ( )

2( ( ) ( )) 3( ( ) ( )) 4 2 ( ) 3 ( ) 4

2 2 ( ) 3( ) 4 2 3 4

4 2( ) 3

4 3 Neglect

f f

f

f

f

t t

t t

t

t

t

t

t

t

J J x t x t J x t

x t x t x t x t dt x t x t dt

x x x x x x x x dt

x x x x dt

x xdt

δ

δ δ

δ δ δ

δ δ δ

δ

Δ = + −

⎡ ⎤ ⎡ ⎤= + + + + − + +⎣ ⎦ ⎣ ⎦

⎡ ⎤⎡ ⎤ ⎡ ⎤= + + + + + − + +⎣ ⎦ ⎣ ⎦⎣ ⎦

⎡ ⎤= + +⎣ ⎦

= +

∫ ∫

∫

∫

∫

Method - 1

( )ing the higher order term

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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[ ]

0

0

0

2

2

2 ( ) 3 ( ) 4 ( )

2 3 4

(Variation of integral = Integral of variation)

4 3

f

f

f

t

t

t

t

t

t

JJ xx

x t x t dt x tx

x x x dtx

x xdt

δ δ

δ

δ

δ

∂⎛ ⎞= ⎜ ⎟∂⎝ ⎠⎡ ⎤∂ ⎡ ⎤= + +⎢ ⎥⎣ ⎦∂ ⎢ ⎥⎣ ⎦

⎡ ⎤∂ ⎡ ⎤= + +⎢ ⎥⎣ ⎦∂⎢ ⎥⎣ ⎦

= +

∫

∫

∫

Method - 2 :

Solution

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Boundary Conditions

Fixed End Point Problems

Free End Point Problems• Completely free

• May be required to lie on a curve

( )( )

0 0, ( ) : Specified

, ( ) : Specifiedf f

t x t

t x tt

( )x t

( )0 0, ( )t x t

( ), ( )f ft x t

( )tη

t

( )x t

( )0 0, ( )t x t

( )tη

( ), ( )f ft x t

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Optimum of a Functional( )

( ) ( ) ( )

( ) ( ) ( )

*

*

* *

A functional is said to have a relative optimum at , if 0

such that for all functions which satisfy ,

the increment of has the "same sign".

1) If 0, then is a r

x t

x t x t x t

J

J J x J x J x

ε

ε

∃ >

∈Ω − <

Δ = − ≥

( ) ( ) ( )

( )

* *

*

elative (local) "Minimum".

2) If 0, then is a relative (local) "Maximum".

Note: If the above relationships are satisfied for arbitrarily large 0,

then is a "global optimum".

J J x J x J x

J x

ε

Δ = − ≤

>

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Optimum of a Functional

a b t

( )* : Optimum pathx t

( )x t

( ) ( ) ( )*Variation: x t x t x tδ = −*,x x

( )x tδ

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Fundamental Theorem of Calculus of Variations

( )

( )

*

*

For to be a candidate for optimum, the following conditions hold good:

1) Necessary Condition: ( ), ( ) 0, admissible ( )

2) Sufficiency Condition:

x t

J x t x t x tδ δ δ= ∀

2

2

0 (for minimum) 0 (for maximum)

JJ

δ

δ

>

<

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Fundamental Lemma

( )

( ) ( )

( )

( )

f

0

t

t

0

0

If for every continuous function

0

where the variation is continuous in , ,

then

0 ,

f

f

g t

g t x t dt

x t t t t

g t t t t

δ

δ

=

⎡ ⎤∈ ⎣ ⎦

⎡ ⎤= ∀ ∈⎣ ⎦

∫

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Necessary condition of optimalityProblem: ( ) ( ) ( )

0

0

Optimize , , by appropriate selection of .

Note: , are fixed.

ft

t

f

J L x t x t t dt x t

t t

⎡ ⎤= ⎣ ⎦∫

Solution: ( )Make sure 0 for arbitrary J x tδ δ=

1) Euler – Lagrange (E-L) Equation

2) Transversality (Boundary) Condition

0L d Lx dt x∂ ∂⎛ ⎞− =⎜ ⎟∂ ∂⎝ ⎠

0

0ft

t

L xxδ∂⎡ ⎤ =⎢ ⎥∂⎣ ⎦

Note: Part of this equation mayalready be satisfied by problem formulation

Necessary Conditions:

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Proof:Hint: Use the necessary condition of optimality from the Fundamental theorem

( )( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ){ }0 0

0 0

*0

*

* * *

* *

Let , [ , ] : Optimumsolution

( ) : Some adjacent solution

Then

, , , ,

, , , ,

f f

f f

f

t t

t t

t t

t tL

x t t t t

x t x t

J J J L x t x t t dt L x t x t t dt

L x t x t t L x t x t t dt L dt

δ

Δ

∈

+

⎡ ⎤⎡ ⎤Δ = − = −⎣ ⎦ ⎣ ⎦

⎡ ⎤⎡ ⎤= − = Δ⎣ ⎦ ⎣ ⎦

∫ ∫

∫ ∫

a b t

( )* : Optimum pathx t

( )x t

( ) ( ) ( )*Variation: x t x t x tδ = −*,x x

( )x tδ

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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( ) ( )

* * * *

* * * *

, ,

However at every point ,

, , , ,

Assumption: is continuous and smooth in both and .

Then, in the limit,

f fx x x x

t

L L x x x x t L x t x t t

L Lx x HOTx x

L x xL LL L xx x

δ δ

δ δ

δ δ

⎡ ⎤ ⎡ ⎤Δ = + + −⎣ ⎦ ⎣ ⎦∂ ∂

= + +∂ ∂

∂ ∂Δ → = +

∂ ∂. In that case,xδ⎡ ⎤

⎢ ⎥⎣ ⎦

Proof

0

ft

t

L LJ J x x dtx x

δ δ δ∂ ∂⎡ ⎤Δ → = +⎢ ⎥∂ ∂⎣ ⎦∫

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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0 0

0 0

0

( )

( ) ( )

f f

ff f

o

ff

o

t t

t t

tt t

t tt

tt

tt

L L d xx dt dtx x dt

L d x d L d xdt dt dtx dt dt x dt

L d Lx xdtx dt x

δδ

δ δ

δ δ

∂ ⎡ ∂ ⎤⎡ ⎤ ⎛ ⎞= ⎜ ⎟⎢ ⎥⎢ ⎥∂ ∂⎣ ⎦ ⎝ ⎠⎣ ⎦

⎡ ⎤∂ ⎡ ∂ ⎤⎛ ⎞ ⎛ ⎞ ⎡ ⎤= −⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥∂ ∂⎝ ⎠ ⎝ ⎠ ⎣ ⎦⎣ ⎦⎢ ⎥⎣ ⎦

⎡ ∂ ⎤ ⎡ ∂ ⎤⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

∫ ∫

∫ ∫ ∫

∫

ProofHowever,

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Proof

0

0 0

0

0 (Necessary condition of optimali

f

f ff

o

f f

o

t

t

t tt

t tt

t t

t t

L LJ x x dtx x

L L d Lxdt x xdtx x dt x

L d L Lxdt xx dt x x

δ δ δ

δ δ δ

δ δ

∂ ∂⎡ ⎤= +⎢ ⎥∂ ∂⎣ ⎦

⎡ ⎤ ⎡ ⎤∂ ∂ ∂⎡ ⎤ ⎛ ⎞ ⎛ ⎞= + −⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎢ ⎥∂ ∂ ∂⎣ ⎦ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤∂ ∂ ∂⎛ ⎞ ⎛ ⎞= − +⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

=

∫

∫ ∫

∫ty)

Hence,

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Necessary Conditions

* Condition (1) must be satisfied regardless of the end condition.* Part of second equation may already be satisfied by the problem of specification. i.e. the amount of extra information contai

Note :

ned by thisequation varies with the boundary conditions specified.

1. 0 (Euler-Lagrange Equation)

2. 0 (Transversality Condition)f

o

t

t

L d Lx dt x

L xx

δ

∂ ∂⎛ ⎞ ⎛ ⎞− =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

⎡ ∂ ⎤⎛ ⎞ =⎜ ⎟⎢ ⎥∂⎝ ⎠⎣ ⎦

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

22

Example – 1Problem: ( ) ( ) ( )

12

0

Minimize with 0 2, 1 3J x x dt x x= + = =∫

Solution:

1) E-L Equation:

( )2L x x= +

( )2

1 210 1 2 0,2 4

L d L tx x x t c t cx dt x∂ ∂⎛ ⎞− = ⇒ − = = ⇒ = + +⎜ ⎟∂ ∂⎝ ⎠

2) Boundary condition: ( ) 20 2x c= =

( )

( )

1

1

2

11 2 341 314 4

3Hence, 24 4

x c

c

t tx t

= + + =

= − =

= + +0

Transversality condition is automatically satisfied, since

0fx xδ δ= =

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

23

Example – 2Problem: ( ) ( )

12

0

Minimize with 0 2,J x x dt x= + =∫

Solution:

1) E-L Equation:

( )2L x x= +

( )2

1 210 1 2 0,2 4

L d L tx x x t c t cx dt x∂ ∂⎛ ⎞− = ⇒ − = = ⇒ = + +⎜ ⎟∂ ∂⎝ ⎠

2) Boundary condition: ( ) 20 2x c= =

( )

( )

1 111

2

0, 0 0

12 0,2 2

Hence, 24 2

f f

f

f

f ft t

ft

t

L Lx xx x

tx c c

t tx t

δ δ

==

∂ ∂= ⇒ = ≠

∂ ∂

= + = ⇒ = −

= − +

∵

( )1 : Freex

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Transversality Condition

0 0

0ff tt

t t

L Lx L x tx xδ δ⎡ ⎤∂ ∂⎡ ⎤ ⎧ ⎫+ − =⎨ ⎬⎢ ⎥⎢ ⎥∂ ∂⎣ ⎦ ⎩ ⎭⎣ ⎦

Special Cases:

( ) ( )0 01) Fixed End Points: , and , are fixed.

No additional information!f ft x t x

0

02) and are fixed (free initial and final states)

0f

f

t

t

t t

L xxδ∂⎡ ⎤ =⎢ ⎥∂⎣ ⎦

General condition:

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Transversality ConditionSpecial Cases:

0 03) , are fixed (free final time, free final state)

0

f f

f ft t

t x

L Lx L x tx x

δ δ∂ ∂⎧ ⎫+ − =⎨ ⎬∂ ∂⎩ ⎭

0 04) , and are fixed (free final time)

0f

f

t

t x x

LL xx∂⎧ ⎫− =⎨ ⎬∂⎩ ⎭

0 05) , and are fixed (free final state)

0

f

f

t

t x t

Lx∂

=∂

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

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Transversality ConditionSpecial Cases:

( ) ( ) ( )0 06) , is fixed; , is constrained to lie on a given curve

0. However,

Hence, the transversality condition is

f ff

f

f f

f f f f f ft tt

ft

t x t x t

L L dx L x t x t tx x dt

L LL xx x

η

ηδ δ δ δ η δ

η

∂ ∂⎧ ⎫+ − = = =⎨ ⎬∂ ∂⎩ ⎭

∂ ∂+ −

∂ ∂( )0 0

f

f ft

t tδ δ⎡ ⎤⎧ ⎫⎢ ⎥ = ≠⎨ ⎬

⎩ ⎭⎢ ⎥⎣ ⎦

( ) 0ft

LL xx

η ∂⎧ ⎫+ − =⎨ ⎬∂⎩ ⎭Finally:

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

27

ExampleProblem: ( ) ( ) ( ) ( )

12

0

Minimize 1 with 0 0 and , lie on 5 15f fJ x dt x t x y t t= + = = − +∫Solution:

1) E-L Equation:

21L x= +

2

0

0 1 0

2

L d Lx dt x

d xdt x

ddt

∂ ∂⎛ ⎞− =⎜ ⎟∂ ∂⎝ ⎠∂⎛ ⎞− + =⎜ ⎟∂⎝ ⎠

2x

( )

2

2

22

2

212 10 0

11

2 1

x xx x xx

xx

x

+ −⎛ ⎞ += ⇒ =⎜ ⎟⎜ ⎟ ++⎝ ⎠

+( ) 22x x x− ( ) 0=

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

28

Example

1) E-L Equation: ( ) 1 20 0L d L x x t c t cx dt x∂ ∂⎛ ⎞− = ⇒ = ⇒ = +⎜ ⎟∂ ∂⎝ ⎠

2) Boundary condition:

( ) ( )

( )

( )

2 1

1 1

(1) 0 0

(2) 0

5 1 0 5 1 0 1/5

Hence, /5 To find , / 5 5 15 75/ 26

ft

f

f f f f

x c x t c t

LL y xx

x c c

x t tt t t t

= = ⇒ =

∂⎡ ⎤+ − =⎢ ⎥∂⎣ ⎦

− + = ⇒ − + = ⇒ =

=

= − + ⇒ =

Variational Problems in Multiple Dimensions: Without Constraints

Dr. Radhakant PadhiAsst. Professor

Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

30

Multiple Dimension Problems without constraints

Problem: ( ) ( ) ( )0

Optimize , , by appropriate selection of .ft

t

J L X t X t t dt X t⎡ ⎤= ⎣ ⎦∫

Solution: ( )Make sure 0 for arbitrary J X tδ δ=

1) Euler – Lagrange (E-L) Equation

2) Transversality (Boundary) Condition

0L d LX dt X∂ ∂⎛ ⎞− =⎜ ⎟∂ ∂⎝ ⎠

Necessary Conditions:

[ ]1 2where TnX x x x

00

0f ft tT

T

tt

L LX L X tX X

δ δ⎡ ⎤ ⎡ ⎤⎧ ⎫∂ ∂⎛ ⎞ ⎛ ⎞+ − =⎢ ⎥ ⎨ ⎬⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎩ ⎭⎢ ⎥ ⎣ ⎦⎣ ⎦

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

31

Variational Problems with Constraints

( )

( )

[ ] [ ]

0

1 2 1 2

Optimize : = , ,

Subject to : , , 0

where

,

ft

t

T Tn n

J L X X t dt

X X t

X x x x ϕ ϕ ϕ

Φ =

Φ

∫

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

32

( ) ( )

( )0

1

*

Lagrange's Existence Theorem:

( ) : The above constrained optimizationproblem leads to the same solution as the followingunconstrained cost functional

= , , , ,

Let , ,

f

n

tT

t

t

J L X X t X X t dt

L X X t L

λ

λ

×∃

⎡ ⎤+ Φ⎣ ⎦

=

∫

( ) ( ), , , ,TX X t X X tλ+ Φ

Variational Problems with Constraints

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

33

* *

* *

Necessary Conditions of Optimality:(1) E-L Equations:

(a) 0 ( n equations)

(b)

L d LX dt X

L d Ldtλ λ

⎡ ⎤∂ ∂− =⎢ ⎥∂ ∂⎣ ⎦

⎡ ⎤∂ ∂− ⎢ ⎥∂ ∂⎣ ⎦

**

0 ( equations)

Note: 0 as there is no term in

n

L Lλλ

=

⎛ ⎞∂=⎜ ⎟∂⎝ ⎠

Variational Problems with Constraints

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

34

* **

*

(2) Transversality Conditions:

(a) 0

(b)

f f

oo

t tTT

tt

L LX L X tX X

L

δ δ

λ

⎡ ⎤ ⎡ ⎤⎧ ⎫⎛ ⎞ ⎛ ⎞∂ ∂⎪ ⎪⎢ ⎥ + − =⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟∂ ∂⎢ ⎥ ⎪ ⎪⎢ ⎥⎝ ⎠ ⎝ ⎠⎩ ⎭⎣ ⎦⎣ ⎦

⎛ ⎞∂⎜ ⎟∂⎝ ⎠

**

f

o

tT

T

t

LLδλ λλ

⎡ ⎤ ⎛ ⎞∂⎢ ⎥ + − ⎜ ⎟⎢ ⎥ ∂⎝ ⎠⎣ ⎦0;

f

o

t

t

tδ⎡ ⎤⎧ ⎫⎪ ⎪⎢ ⎥ =⎨ ⎬⎢ ⎥⎪ ⎪⎩ ⎭⎣ ⎦

Variational Problems with Constraints

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

35

( )

( ) ( )

* *

*

0 0

* **

E-L Equations:

1) (a) 0

(b) , , 0 (same constraint equation)

2)Transversality Conditions: , fixed, , free

(a) f

f f

TT

ft

L d LX dt X

L X X t

t X t X

L LX L XX

λ

δ

⎛ ⎞ ⎛ ⎞∂ ∂− =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

⎛ ⎞∂= Φ =⎜ ⎟∂⎝ ⎠

⎛ ⎞∂ ∂+ −⎜ ⎟∂ ∂⎝ ⎠

*

*

0 ( equations)

(b) 0 However is free 0

so 0 ( 1 equation)

f

f

f

f

t

t f f f

t

t nX

L t t t

L

δ

δ δ

⎡ ⎤⎛ ⎞=⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

= ⇒ ≠

=

Variational Problems with Constraints

Varaibles: 1 ( ) ( ) ( )

Boundary Conditions : 1f

n nX t

n n

λ+ +

+ +

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

36

Constraint Equations

Nonholonomic constraints

Isoperimetric constraints

( ), , 0X X tΦ =

( )0

, ,

One way to get rid of Isoperimetric constraints is to convertthem into Nonholonomic constraints.

ft

t

q X X t dt k=∫

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

37

Isoperimetric Constraints( )

0

1

1

1 1 0

1 1 0

1 0

1

Define: , ,

Then

( ) ( )

Choose one of ( ) or ( ) and fix the other

Let ( ) 0 ( )

f

n

t

nt

n f n

n f n

n

n f

x q X X t

x dt k

x t x t k

x t x t

x tx t k

+

+

+ +

+ +

+

+

=

=

− =

=

=

∫

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

38

( )1

1 0

1

The following additional non-holonomicconstraint is introduced:

, ,

with boundary conditions: ( ) 0 ( )

n

n

n f

x q X X t

x tx t

+

+

+

=

=

Summary :

The original problem is augmentedwith this information and solved.

k=

Isoperimetric Constraints

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

39

References

T. F. Elbert, Estimation and Control Systems, Von Nostard Reinhold, 1984.

D. S. Naidu, Optimal Control Systems, CRC Press, 2002.

ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore

40